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Local class field theory. (English) Zbl 0604.12014
Oxford Mathematical Monographs. New York: Oxford University Press; Oxford: Clarendon Press. VIII, 155 p. (1986).
[See also the author’s Japanese book with the same title. For the Russian translation (1983) of this cf. Zbl 0566.12005.] This book is a very interesting introduction to class field theory for local fields. The main feature is the fundamental use of the theory of formal groups in the proofs. As formal groups are now an important tool in the subject, it is a good idea to teach them early to our students. It is perhaps the best way to prepare them to methods from algebraic geometry. Now let’s present the book more precisely. First of all, a preface summarizes the history of the subject from it birth in the thirties. The first part (chap. I, II, III) introduces the main objects to be studied: valuations, ramification, units, different, discriminants, representations of elements as sums of power series in an uniformizing parameter $\dots$ The second part is the kernel of the book. After introducing the formal groups it studies the extensions generated by their torsion points. One can easily describe their Galois groups as well as the groups of relative norms. The chapter VI gives the classical theorems: introduction of the reciprocity law and its “functorialities”. The existence theorem is proved in chapter VII. Of course, the upper numeration of ramification groups is studied and the Hasse-Arf theorem is proved. In chapter VIII, one finds the explicit formulas of Wiles. An appendix gives the definitions of Galois cohomology groups with accent on $H\sp 2$ (Brauer group) and a method due to {\it M. Hazewinkel} [Adv. Math. 18, 148-181 (1975; Zbl 0312.12022)]. As Hazewinkel said “local class field theory is easy”, but as Iwasawa’s book shows, it is a beautiful theory!
Reviewer: R. Gillard

11S31Class field theory for local fields; $p$-adic formal groups
11-02Research monographs (number theory)
11S15Ramification and extension theory
11S25Galois cohomology for local fields
12G05Galois cohomology
14L05Formal groups, $p$-divisible groups
14F22Brauer groups of schemes